Path Integrals
We have seen two equivalent ways to do quantum mechanics: the Schrödinger picture (wavefunctions evolve) and the Heisenberg picture (operators evolve). Feynman discovered a third approach in 1948: the path integral or sum over histories. It connects quantum mechanics directly to the principle of least action, the deepest link between classical and quantum physics.
The Central Idea
A particle going from A to B doesn't take one path. It takes all paths simultaneously, and each path contributes a phase factor proportional to its classical action.
The Propagator
Probability amplitude
A probability amplitude is a complex number whose squared magnitude gives a probability. Unlike ordinary probabilities, amplitudes can be negative or complex, and they can interfere with each other (adding up or canceling out). This interference is the essence of quantum behavior and is what makes path integrals work.
The key object is the propagator (or kernel) $K(x_f, t_f; x_i, t_i)$: the probability amplitude for a particle starting at position $x_i$ at time $t_i$ to end at position $x_f$ at time $t_f$.
In the Schrödinger picture:
$$\psi(x_f, t_f) = \int K(x_f, t_f; x_i, t_i) \psi(x_i, t_i) \, dx_i$$The propagator contains everything about time evolution.
Feynman's Path Integral
$$K(x_f, t_f; x_i, t_i) = \int \mathcal{D}[x(t)] \, \exp\left(\frac{i}{\hbar}S[x(t)]\right)$$Sum over all paths $x(t)$ from $(x_i, t_i)$ to $(x_f, t_f)$, weighted by $e^{iS/\hbar}$.
What is the action?
The action $S$ is a single number that summarizes an entire trajectory. It is computed by adding up, at every moment along the path, the difference between kinetic and potential energy (this difference is called the Lagrangian $L$). In classical physics, nature "chooses" the path that extremizes the action. In quantum mechanics, every path contributes, weighted by $e^{iS/\hbar}$.
The Lagrangian
The Lagrangian $L(x, \dot{x}, t)$ equals kinetic energy minus potential energy: $L = T - V$. While the Hamiltonian (total energy $H = T + V$) governs the Schrödinger equation, the Lagrangian is the natural quantity in the path integral formulation. The dot notation $\dot{x}$ means the velocity (how fast $x$ changes with time).
Here $S[x(t)]$ is the classical action along path $x(t)$:
$$S[x(t)] = \int_{t_i}^{t_f} L(x, \dot{x}, t) \, dt$$The symbol $\int \mathcal{D}[x(t)]$ means "integrate over all possible paths," an infinite-dimensional integral.
Making It Precise: Time-Slicing
How do we integrate over "all paths"? Discretize time into $N$ steps:
$$t_i = t_0 < t_1 < t_2 < \cdots < t_N = t_f$$with spacing $\epsilon = (t_f - t_i)/N$. A "path" is then a sequence of positions $x_0, x_1, \ldots, x_N$.
Discretized Path Integral
$$K = \lim_{N \to \infty} \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2} \int dx_1 \cdots dx_{N-1} \exp\left(\frac{i}{\hbar}\sum_{j=0}^{N-1} L_j \epsilon\right)$$At each time slice, we integrate over all possible positions. In the $N \to \infty$ limit, we sum over all continuous paths.
Example: The Free Particle
Gaussian integrals
A Gaussian integral is an integral over a bell-curve-shaped function, $\int e^{-ax^2} dx$. These integrals have exact closed-form solutions and appear constantly in physics. In the path integral, each time-slice integral turns out to be Gaussian when the Lagrangian is quadratic (proportional to $x^2$ or $\dot{x}^2$), which is why certain path integrals can be computed exactly.
For a free particle, $L = \frac{1}{2}m\dot{x}^2$. The path integral is a product of Gaussian integrals and can be evaluated exactly:
Free Particle Propagator
$$K(x_f, t_f; x_i, t_i) = \sqrt{\frac{m}{2\pi i\hbar (t_f - t_i)}} \exp\left(\frac{im(x_f - x_i)^2}{2\hbar(t_f - t_i)}\right)$$This matches exactly what we get from solving the Schrödinger equation. The path integral is equivalent to standard quantum mechanics.
The Classical Limit
How does classical mechanics emerge from summing over all paths? The key is the stationary phase approximation.
Stationary phase approximation
When you add up many oscillating contributions $e^{i\theta}$ with rapidly varying phases $\theta$, they tend to cancel each other out. The only region that survives is where the phase changes slowly (is "stationary"), because nearby contributions point in the same direction and reinforce each other. This is how a single classical path emerges from the infinity of quantum paths.
Phase Interference
Each path contributes with phase $e^{iS/\hbar}$. For typical actions $S \gg \hbar$, nearby paths have vastly different phases and cancel out. The only paths that survive are those where the phase is stationary:
$$\frac{\delta S}{\delta x(t)} = 0$$Functional derivative
The symbol $\delta S / \delta x(t)$ is a functional derivative. An ordinary derivative asks "how does a function change when I nudge its input?" A functional derivative asks "how does a number (the action $S$) change when I slightly deform an entire function (the path $x(t)$)?" Setting this to zero picks out the path where the action is at an extremum, which is the classical trajectory.
This is exactly the classical equation of motion! The classical path is where contributions add constructively.
The Semiclassical Expansion
Expanding around the classical path $x_{cl}(t)$:
$$x(t) = x_{cl}(t) + y(t)$$The action becomes:
$$S[x] = S[x_{cl}] + \frac{1}{2}\int y \cdot \frac{\delta^2 S}{\delta x^2}\bigg|_{cl} \cdot y \, dt + \cdots$$The leading term is Gaussian in $y$ and gives quantum corrections to the classical result.
Why Path Integrals Matter
1. Quantum Field Theory
Path integrals are the natural language for QFT. Instead of summing over particle paths, we sum over field configurations:
$$Z = \int \mathcal{D}[\phi] \, e^{iS[\phi]/\hbar}$$The partition function
The partition function $Z$ is a master quantity that encodes all the physics of a system. From it, you can extract energies, transition probabilities, correlation functions, and thermodynamic properties. It originated in statistical mechanics (where it counts how states are distributed across energies) and plays the same central organizational role in quantum field theory.
This is the "partition function" from which all physics is derived.
2. Non-Perturbative Effects
What are instantons?
Instantons are special field configurations that describe quantum tunneling between different vacuum states. Unlike ordinary perturbation theory, which considers small fluctuations, instantons represent large, sudden rearrangements of the field. They are invisible to perturbative methods but are naturally captured by the path integral, which sums over all configurations including these rare but important ones.
Some phenomena (instantons, tunneling between vacua) involve paths far from the classical solution. Path integrals capture these naturally.
3. Lattice Gauge Theory
What is lattice gauge theory?
Lattice gauge theory replaces continuous spacetime with a discrete grid (lattice) of points. The path integral, which is normally an impossible infinite-dimensional integral, becomes a very large but finite sum that a computer can handle. This is the primary tool for calculating the properties of the strong nuclear force (QCD) from first principles.
Discretizing spacetime, path integrals become ordinary (high-dimensional) integrals that can be computed numerically. This is how we calculate QCD from first principles.
4. Statistical Mechanics Connection
What is a Wick rotation?
A Wick rotation is a mathematical trick where you replace real time $t$ with imaginary time $\tau = it$. This transforms the oscillating quantum phase factor $e^{iS/\hbar}$ into a decaying exponential $e^{-S_E/\hbar}$, which is much better behaved mathematically. The result looks exactly like the formulas of statistical mechanics, revealing a deep connection: quantum mechanics in $d$ spatial dimensions is mathematically equivalent to statistical mechanics in $d+1$ dimensions.
Replace $it \to \tau$ (Wick rotation to imaginary time), and the quantum path integral becomes a statistical mechanics partition function:
$$Z = \int \mathcal{D}[x(\tau)] \, e^{-S_E[x]/\hbar}$$Quantum mechanics in $d$ dimensions is equivalent to statistical mechanics in $d+1$ dimensions!
Deep Connections
The path integral reveals that quantum amplitudes are determined by the classical action, that quantum corrections come from fluctuations around classical paths, and that quantum mechanics and statistical mechanics are two facets of the same mathematics.
The Action as Central Object
We began our study of physics with the principle of least action. Now we see it at the heart of quantum mechanics:
| Classical | Quantum | |
|---|---|---|
| Principle | $\delta S = 0$ | Sum over all $S$ with phase $e^{iS/\hbar}$ |
| Paths | One path (extremizes $S$) | All paths contribute |
| Why classical works | (assumed) | Other paths interfere destructively |
Quantum mechanics does not abandon the action; it democratizes it. Every path gets a vote, weighted by its action. Classical physics emerges when one path's vote dominates.
Key Insights
- In quantum mechanics, particles take all paths between two points
- Each path contributes amplitude $e^{iS/\hbar}$, where $S$ is the classical action
- The propagator is the sum (integral) over all these contributions
- Classical mechanics emerges because paths near $\delta S = 0$ interfere constructively
- Path integrals are the foundation of quantum field theory
- The Wick rotation connects quantum mechanics to statistical mechanics
- The action, introduced for classical physics, is equally central to quantum mechanics
Looking Ahead
You've now completed the foundations of quantum mechanics. The path integral formulation will be essential as we move to:
- Quantum Field Theory: Particles as excitations of fields
- The Standard Model: The quantum theory of all known particles
- Quantum Gravity: The open frontier
The journey continues in Phase 3.
- In Feynman's path integral formulation, a quantum particle takes all possible paths simultaneously, each weighted by a phase factor $e^{iS/\hbar}$ determined by the classical action.
- Classical mechanics emerges because paths near the classical trajectory interfere constructively, while all other paths cancel out through destructive interference.
- Path integrals are the natural language of quantum field theory and provide the foundation for the Standard Model of particle physics.
- The Wick rotation ($t \to -i\tau$) reveals a deep mathematical equivalence between quantum mechanics and statistical mechanics.